Teach Me Maths

Translations and Column Vectors

On this page

How to read a column vector, translate points and shapes on a grid, find the vector that maps one shape onto another, work backwards to where a point started, and describe a translation fully.

What a translation is

A translation slides a shape to a new position. Every point of the shape moves the same distance in the same direction. The image is the same size and shape as the original, and the same way round: it is not turned and not flipped.

In the exam you are usually asked to translate a shape on a grid by a column vector, for 1 or 2 marks, or to describe fully the single transformation that maps one shape onto another, for 2 marks. You may also be asked for the numbers in a vector, or for where a point started.

Here every corner of triangle A moves 5 squares right and 3 squares down to make triangle B. The arrows are all the same length and point the same way.

Column vectors

A translation is written as a column vector: two numbers, one above the other, in tall brackets. The top number is the move across. The bottom number is the move up or down.

  • Top number positive: move right. Top number negative: move left.
  • Bottom number positive: move up. Bottom number negative: move down.
  • A 0 means no move that way: \begin{pmatrix}0\\4\end{pmatrix} is 4 squares up.

So \begin{pmatrix}3\\-2\end{pmatrix} means 3 squares right and 2 squares down. Go across first, then up or down.

To translate a point by the vector \begin{pmatrix}a\\b\end{pmatrix}, add the top number to the x-coordinate and the bottom number to the y-coordinate. The point (x,y) moves to

(x+a,\ y+b)

where a is the top number and b is the bottom number.

Worked example 1: Translating a point

The point A(2,5) is translated by the vector \begin{pmatrix}-6\\-3\end{pmatrix}. Write down the coordinates of the image of A.

  1. The top number is -6, so move 6 squares left: 2-6=-4

  2. Your turn. What is the y-coordinate of the image?

    The bottom number is -3, so move 3 squares down: 5-3=2

  3. Answer: (-4,2)

Translating a shape

To translate a shape, move each corner, or vertex, by the vector, one at a time. Then join the new corners with a ruler, in the same order as the original.

Count the squares from a corner, not from the middle of a side. Check your image: it is the same size and shape as the original, and the same way round.

Tracing paper helps, and you can ask for it in the exam. Trace the shape, slide the paper by the vector without turning it, and see where the shape lands.

Worked example 2: Translating a triangle

Triangle PQR has vertices P(-4,1), Q(-1,2) and R(-3,4). Translate triangle PQR by the vector \begin{pmatrix}5\\-4\end{pmatrix}

  1. The vector means 5 right and 4 down. Add 5 to each x-coordinate and subtract 4 from each y-coordinate.
  2. P(-4,1) moves to P′(-4+5,\ 1-4)=(1,-3)
  3. Q(-1,2) moves to Q′(-1+5,\ 2-4)=(4,-2)

  4. Your turn. Write down the coordinates of R′.

    Write your answer as (x, y)

    R(-3,4) moves to R′(-3+5,\ 4-4)=(2,0)

  5. Answer: P′(1,-3), Q′(4,-2) and R′(2,0). Join them with a ruler.

Finding the vector

To find the vector of a translation, pick one corner and the matching corner of its image. Count the squares across, then up or down. Or subtract the coordinates:

\text{top} = \text{new } x - \text{old } x \qquad \text{bottom} = \text{new } y - \text{old } y

Direction matters. If the vector from shape A to shape B is \begin{pmatrix}4\\-1\end{pmatrix}, then the vector from B back to A is \begin{pmatrix}-4\\1\end{pmatrix}: both signs change. Always start from the shape that is being moved.

Working backwards: if you know the image and the vector, go the opposite way. Subtract the top number from the x-coordinate and the bottom number from the y-coordinate.

Worked example 3: Finding the numbers in a vector

Point M(3,-2) is translated to point N(-1,4). The translation is the vector \begin{pmatrix}c\\d\end{pmatrix}. Find the value of c and the value of d.

  1. Top number: new x minus old x. c=-1-3=-4
  2. Check by counting: from 3 to -1 is 4 squares left, so c is negative.

  3. Your turn. Find the value of d.

    d=4-(-2)=6, which is 6 squares up

  4. Answer: c=-4 and d=6

Worked example 4: Finding where a point started

Point K is translated by the vector \begin{pmatrix}-5\\2\end{pmatrix}. Its image is K′(-1,4). Find the coordinates of K.

  1. The vector moved K 5 left and 2 up. To get from K′ back to K, go the opposite way: 5 right and 2 down.
  2. x-coordinate of K: -1+5=4

  3. Your turn. Write down the coordinates of K.

    Write your answer as (x, y)

    y-coordinate of K: 4-2=2, so K is (4,2)

  4. Check: (4-5,\ 2+2)=(-1,4), which is K′.
  5. Answer: K(4,2)

Describing a translation fully

First decide what kind of transformation it is. If the image is the same size and shape and the same way round, it is a translation. If it has been turned, it is a rotation; if it faces the other way, it is a reflection; if it is a different size, it is an enlargement.

To describe fully a translation, give two things: the word translation and the column vector. For example: translation by the vector \begin{pmatrix}-2\\6\end{pmatrix}

Write translation, not moved, shift or transition. Give the vector in tall brackets, not in words such as 2 left and 6 up, not as a coordinate such as (-2,6), and with no fraction line. Give only one transformation.

Worked example 5: Describing a translation from a grid

Describe fully the single transformation that maps shape A onto shape B.

  1. B is the same size and shape as A, and the same way round. It has not been turned or flipped, so the transformation is a translation.
  2. Pick one corner of A and the matching corner of B: the top left corner of A, (2,4), moves to the top left corner of B, (-4,-1).
  3. Across: from 2 to -4 is 6 squares left, so the top number is -6.

  4. Your turn. What is the bottom number of the vector?

    Down: from 4 to -1 is 5 squares down, so the bottom number is -5.

  5. Check with another corner: (2,2) moves to (-4,-3), also 6 left and 5 down.
  6. Answer: translation by the vector \begin{pmatrix}-6\\-5\end{pmatrix}

Worked example 6: Counting the right way

Triangle P has vertices (2,-3), (5,-3) and (2,1). Triangle Q has vertices (-5,1), (-2,1) and (-5,5). Describe fully the single transformation that maps triangle Q onto triangle P.

  1. Q is mapped onto P, so Q is the shape being moved. Count from Q to P.
  2. Q and P are the same size and shape, and the same way round, so the transformation is a translation. The corner (-5,1) of Q matches the corner (2,-3) of P: each is the bottom left corner.
  3. Top number: 2-(-5)=7, which is 7 squares right

  4. Your turn. What is the bottom number of the vector?

    Bottom number: -3-1=-4, which is 4 squares down

  5. Check with another corner: (-2,1) moves to (5,-3), also 7 right and 4 down.
  6. Answer: translation by the vector \begin{pmatrix}7\\-4\end{pmatrix}

Common mistakes

  • Swapping the two numbers. The top number is across and the bottom number is up or down.
  • Getting a sign wrong. Left and down are negative; right and up are positive.
  • Going the wrong way. Count from the shape being moved to its image, or both signs come out wrong.
  • Not saying translation. Use the word translation, not moved, shift or transition.
  • Writing the vector wrongly. Write it in tall brackets with no line between the numbers, not in words, as a coordinate or as a fraction.
  • Giving more than one transformation. Describe one translation only, or you can lose every mark.

Practice questions

More on this topic: Translations and Column Vectors worksheet with full solutions